We consider the two-dimensional mean field equation of the equilibrium turbulence with variable intensities and Dirichlet boundary condition on a pierced domain −Δu=λ1 [Formula presented] −λ2τ [Formula presented] in Ωϵ=Ω∖⋃i=1mB(ξi,ϵi)¯u=0on ∂Ωϵ, where B(ξi,ϵi) is a ball centered at ξi∈Ω with radius ϵi, τ is a positive parameter and V1,V2>0 are smooth potentials. When λ1>8πm1 and λ2τ2>8π(m−m1) with m1∈0,1,…,m, there exist radii ϵ1,…,ϵm small enough such that the problem has a solution which blows-up positively and negatively at the points ξ1,…,ξmjavax.xml.bind.JAXBElement@4bea4a04 and ξmjavax.xml.bind.JAXBElement@8c2344a+1,…,ξm, respectively, as the radii approach zero.
Esposito, P., Figueroa, P., Pistoia, A. (2020). On the mean field equation with variable intensities on pierced domains. NONLINEAR ANALYSIS, 190, 111597 [10.1016/j.na.2019.111597].